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Quasi-bipartite graph

Quasi-bipartite graph is a science topic covered in the lgStudy science library. This page brings together a partial reference excerpt, illustrations, worked examples, real-world applications and a short study plan, so you can understand Quasi-bipartite graph rather than just read about it. In short: In the mathematical field of graph theory, an instance of the Steiner tree problem (consisting of an undirected graph G and a set R of terminal vertices that must be connected to each other) is said to be quasi-bipartite if the non-terminal vertices in G form an independent set, i.e. if every edge is incident on at least one terminal. This generalizes the concept of a bipartite graph: if G is bipartite, and R is the…

Quasi-bipartite graph — main illustration
Quasi-bipartite graph — illustration

Key takeaways

  • Quasi-bipartite graph belongs to science; place it in that map before memorising details.
  • Learn the definition first, then one example that makes the definition concrete.
  • Connect Quasi-bipartite graph to a quantity you can measure, compute or draw — that is where exam questions come from.
  • Reproduce the core statement of Quasi-bipartite graph from memory before moving on to harder problems.

Reference excerpt

In the mathematical field of graph theory, an instance of the Steiner tree problem (consisting of an undirected graph G and a set R of terminal vertices that must be connected to each other) is said to be quasi-bipartite if the non-terminal vertices in G form an independent set, i.e. if every edge is incident on at least one terminal. This generalizes the concept of a bipartite graph: if G is bipartite, and R is the set of vertices on one side of the bipartition, the set R is automatically independent.

This concept was introduced by Rajagopalan and Vazirani who used it to provide a (3/2 + ε) approximation algorithm for the Steiner tree problem on such instances. Subsequently, the ε factor was removed by Rizzi and a 4/3 approximation algorithm was obtained by Chakrabarty et al. The same concept has been used by subsequent authors on the Steiner tree problem, e.g. Robins and Zelikovsky proposed an approximation algorithm for Steiner tree problem which on quasi-bipartite graphs has approximation ratio 1.28. The complexity of Robins and Zelikovsky's algorithm is O(m n2), where m and n are the numbers of terminals and non-terminals in the graph, respectively. In 2012, Goemans et al. gave a 73/60 ≈ 1.217-approximation algorithm for the Steiner tree problem on quasi-bipartite graphs; an algorithm achieving the same approximation factor was previously known for the special case of quasi-bipartite graphs with unit cost edges.

References

Illustrations

Quasi-bipartite graph: The first graph has non-terminal points which are directly connected, and is therefore non-quasi-bipartite. The second has no non-terminal points which are directly connected, and is therefore quasi-bipartite.
The first graph has non-terminal points which are directly connected, and is therefore non-quasi-bipartite. The second has no non-terminal points which are directly connected, and is therefore quasi-bipartite.

Worked examples

Example 1 — a first encounter with Quasi-bipartite graph

Start with the simplest possible case. Write down what Quasi-bipartite graph claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to Quasi-bipartite graph before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about Quasi-bipartite graph ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of Quasi-bipartite graph

In research
Quasi-bipartite graph appears in science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry
Engineering practice reuses Quasi-bipartite graph in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom
Quasi-bipartite graph is common in secondary-school and first-year university syllabi. It links to neighbouring topics Bipartite graphs, Graph families, so understanding it makes those chapters shorter.
In everyday life
Look for Quasi-bipartite graph outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.
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How to study Quasi-bipartite graph in 20 minutes

  1. Read the reference excerpt below once, without taking notes.
  2. Close the page and write down what Quasi-bipartite graph means in your own words.
  3. Compare your version with the excerpt and mark what you missed.
  4. Work through the three examples above with pen and paper.
  5. Explain Quasi-bipartite graph out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is Quasi-bipartite graph in simple terms?

In the mathematical field of graph theory, an instance of the Steiner tree problem (consisting of an undirected graph G and a set R of terminal vertices that must be connected to each other) is said to be quasi-bipartite if the non-terminal vertices in G form an independent set, i.e. if every edge…

Why does Quasi-bipartite graph matter?

Because it connects several science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study Quasi-bipartite graph?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on Quasi-bipartite graph.

Tags

  • Bipartite graphs
  • Graph families

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